散乱数据点多项式插值光顺曲面的构造

为了得到光顺的多项式插值曲面,首先把空间散乱数据点划分为三角形网格,在每个给定数据点处构造C^1连续的分片二次多项式曲面片,针对各数据点的邻接点个数不同,分别利用弯折能量和拉伸能量建立目标函数,极小化目标函数确定插值曲面的未知量,在保持原有的形状特征的同时构造光顺的分片插值曲面,最后用实例说明了文中方法的有效性．     

三角形域上C1连续的四次插值曲面

提出了一种在三角形域上构造C^1曲面的方法，该方法构造的曲面片由4个曲面加权平均产生，在三角形的边界上满足给定的边界曲线和一阶跨界导数．所构造的曲面可看作由一张基本曲面和三张过渡曲面构成．用三条曲线相交于一点且在交点处共面作为约束条件构造基本曲面，在三角形的内部具有较好形状和逼近精度．同边点法相比，文中方法产生的曲面形状更好；且该方法产生的曲面对四次多项式曲面是精确的，因而比Nielson的点边方法具有更高的插值精度．     

散乱数据点的三次多项式插值

用分片三次多项式曲面对散乱分布数据点插值的方法把给定区域划分成三角形网格，在每个三角形上构造一个三次多项式曲面片，整体的Ｃ１曲面由各三角形上的曲面片拼合而成．讨论了整体Ｃ１曲面需满足的条件组成的方程组的性质，并给出了求解方程组的方法．插值方法的多项式准确集包括所有三次和小于三次的多项式．     

一个构造插值曲面方法中存在的错误

一、引言对N个任意分布的空间数据点{(x_i,y_i,z_i)}_(i=1)~N,文章[1]中提出了一个构造二元 插值曲面的方法。做法是把给定数据点在XY平 面上划分成三角形网格,在每个数据点p_i=(x_i, y_i,z_i)处构造一个对z_i插值的曲面片f_i(x,y)。 每个三角形上的曲面片由三个顶点处的曲面片加 权平均产生,整体的曲面由所有三角形上的曲面 片拼合而成。方法的具体做法如下:     

在三角域上构造三次多项式插值曲面

为满足矿山地形的拟合、水流深度的绘制等很多特殊工程数据量大、有一定的光顺要求但又不需要曲面过于凸起饱满这一需求,提出一种C1连续的三次多项式插值曲面,同时有针对性地提出一种一阶偏导数估计算法.首先将空间散乱数据点投影到平面后进行三角划分;其次针对每个三角形,在其每条边上构造一个C1连续的三次多项式曲面片,由这3个曲面片加权平均形成该三角形的曲面片;最后将所有三角形上的曲面片拼合成整体曲面.为使生成的曲面尽可能地贴近数据点所建议的形状,在曲面求解过程中将数据点分成内部点和边界点分别估计偏导数.实验结果表明,该算法计算量小、具有良好的局部性,并给出了新曲面的效果.     

三角网格上的代数曲面重建

提出一种用分片代数曲面构造三角曲面片的方法,利用具有公共边的2个三角形区域的4个顶点的函数值以及公共边2个端点的外法向量来构造一个二次曲面V(g)和一个截面V(h),其交V(g,h)即为2个三角曲面片的公共边界曲线.对每个已确定了边界条件的三角片内部进一步划分成3部分,每部分各自定义一个三次代数曲面.这3个三次代数曲面不仅在其交线处光滑拼接,而且分别沿三角形的边界与V(g)光滑拼接,从而构成一个具有GC1连续性的分片代数曲面.对于只属于一个三角片的边界留有一个自由度,可对曲面形状加以控制.     

三角域上C1连续的H-Coons曲面片

在三角形域上利用两类H-Hermite多项式构造C1连续的两种形式的H-Coons曲面片。构造的三角曲面片均含有形状参数  ,调整  的值,可改变曲面的内部形状,而不影响曲面的边界形状。当 0 时,可退化为通常的边-边与边-点方法插值的曲面片。最后,通过实例显示了该文方法的实际效果。     

带形状参数的C1连续的三角Coons曲面片

针对三角形域上超限插值的曲面缺乏形变的特点,利用两类带形状参数的Hermite多项式构造C1连续的两种格式的带形状参数的Coons曲面片。构造的三角曲面片均含有形状参数λ,调整λ的值,可改变曲面的内部形状,而不影响曲面的边界形状;当λ＝0时,可退化为通常的边-边与边-点格式插值的曲面片。最后,实例验证了该方法的有效性。     

三角域上C 1连续的C-Coons曲面片*

在三角形域上利用两类C-Hermite多项式构造C1连续的两种形式的C-Coons曲面片,构造的三角曲面片均含有形状参数α,调整α的值,可改变曲面的内部形状,而不影响曲面的边界形状;当形状参数α趋于0时,可退化为通常的边—边与边—点方法插值的曲面片。最后,通过例子比较显示了该方法的实际效果。     

三角域上C1连续的T-Coons曲面片

在三角形域上利用两类含形状参数的三角Hermite多项式构造C¹连续的两种形式的带形状参数的Coons曲面片。构造的三角曲面片均含有形状参数λ,调整λ的值,可改变曲面的内部形状,而不影响曲面的边界形状。给定的方法是对通常的边－边法和边－点法的扩展。给出的两个实例显示了方法的有效性。     

类螺旋特征测点数据的闭曲面建模方法研究

复杂曲面及海量点云测量数据的曲面建模已成为通用CAD/CAM软件的重要功能;然而,对于复杂的闭曲面建模方法,仍然存在许多技术上的难题,至今尚未能很好的解决,比如,基于海量的测量数据,如何进行闭曲面特征点识别,如何进行区域分割与处理,这一切都使得闭曲面建模过程中很难采用已经成熟的自由曲面建模技术和方法.通过研究异步仿形测量原理以及测量数据类型,针对鞋楦测量形成的空间螺旋线数据特征,提出一种闭曲面建模方法.该方法包括如下步骤:首先对测量点数据处理;并以特征螺旋线数据为基础对曲面进行三角分割;最后,以三角Bezier曲面为基础进行曲面构造,并将各曲面进行拼接、裁剪,形成完整的曲面.采用该方法对鞋楦测量数据的建模实例说明,能够有效地对具有空间螺旋线数据特征的闭曲面进行数据处理、曲面重构,提高了产品建模效率.     

散乱数据曲面造型的凸组合Bezier三角曲面片插值方法

本文讨论了在计算机辅助设计和计算机图形学的散乱数据曲面造型中一种有效的凸组合Ｂｅｚｉｅｒ三角曲面片插值方法。构造曲面的方法是对三角剖分的每一条边,得到一个插值已知条件的５次Ｂｅｚｉｅｒ三角曲面片,它与共此边的相邻５次Ｂｅｚｉｅｒ三角曲面片满足Ｃ＾２连续条件,然后对三角剖分的每一个三角形,将三边对应的Ｂｅｚｉｅｒ三角曲面片作凸组合,使之仍然插值已知条件并满足Ｃ＾２连续条件,从数值例子看,效果是不错的     

An Algorithm for Parametric Quadric Patch Construction

An algorithm for constructing arbitrary parametric quadratic quadric triangles in rational Bézier form is presented. The algorithm does not require the knowledge of the underlying quadric, an important property in view of applying this method for the interpolation of triangulated 3D data points. The algorithm consists of four steps starting with the arbitrary choice of the three corner points and corner weights of the patch, by then constructing a certain triangle and a tetrahedron by means of which the remaining inner control points and weights are obtained guaranteeing the resulting patch to lie on a quadric surface.     

Shape aware normal interpolation for curved surface shading from polyhedral approximation

Independent interpolation of local surface patches and local normal patches is an efficient way for fast rendering of smooth curved surfaces from rough polyhedral meshes. However, the independently interpolating normals may deviate greatly from the analytical normals of local interpolating surfaces, and the normal deviation may cause severe rendering defects when the surface is shaded using the interpolating normals. In this paper we propose two novel normal interpolation schemes along with interpolation of cubic Bézier triangles for rendering curved surfaces from rough triangular meshes. Firstly, the interpolating normal is computed by a Gregory normal patch to each Bézier triangle by a new definition of quadratic normal functions along cubic space curves. Secondly, the interpolating normal is obtained by blending side-vertex normal functions along side-vertex parametric curves of the interpolating Bézier surface. The normal patches by these two methods can not only interpolate given normals at vertices or boundaries of a triangle but also match the shape of the local interpolating surface very well. As a result, more realistic shading results are obtained by either of the two new normal interpolation schemes than by the traditional quadratic normal interpolation method for rendering rough triangular meshes.     

用有理双三次Bezier曲面片混合二次曲面

提出一种二次曲面混合方法，混合曲面由2张有理双三次B6zier曲面片构成，它们之间保持G^2连续，混合曲面与二次曲面间保持G^1连续．给出了混合曲面片控制顶点的显式表示，通过修改2类混合参数可以直观地调节混合方向及混合曲面的形状．另外，混合5个圆锥曲面的例子表明，该方法为多个二次曲面的混合问题提供了有效途径．     

Constructing triangular patch by basic approximation operator plus additional interpolation operator

~~Constructing triangular patch by basic approximation operator plus additional interpolation operator1. Barahill, R. E., Birkhoff, G., Gordon, W. J., Smooth interpolation in triangles, J. Approx. Theory, 1973, 8: 114-128. 2. Gregory, J. A., Smooth interpolation without twist constraints, in Computer Aided Geometric Design (eds. Barn-hill, R. E., Riesenfeld, R. R), New York: Academic Press, 1974, 71-88. 3. Charrot, P., Gregory, J. A., A pentagonal surface patch for comput…     

Nonsplitting Macro Patches for Implicit Cubic Spline Surfaces

Macro patches are important for generating quadric or cubic implicit spline surfaces from the input of a polyhedron. All existing macro patches split the triangular facets of the polyhedron; this paper presents cubic nonsplitting macro patches (NMP) that do not split these facets. The NMP's are based on a necessary and sufficient condition for nonsplitting constructions of implicit cubic spline surfaces. This condition can be satisfied for most practical applications, so the NMP's lead to an efficient and powerful spline surface scheme using implicit cubics. The free parameters in an NMP are set using a new technique for excluding topological anomalies such as extraneous sheets, splits, unwanted holes, self-intersections, and unwanted handles. Each cubic patch obtained by this technique best approximates, in a least-squares sense, a quadric patch from a single algebraic component of a monotone polynomial derived from the input data.     

Smooth closed surfaces with discrete triangular interpolants

Discrete interpolants which involve cross boundary derivatives in an attempt to form C¹ surfaces have the following major problem: Requiring C¹ joins between patches makes sense only if the patch domains are adjacent in the domain space. This makes it impossible to form C¹ closed surfaces, or indeed any surface which contains more connections than can be achieved in the domain.

This paper develops a method of forming smooth closed (or otherwise complexly connected) surfaces from a discrete triangular interpolant by relaxing the C¹ property of an interpolant to ‘Visually C¹”.

The only constraint on the scheme is that the data to be interpolated define a unique tangent plane at each vertex where several triangles meet. Then each patch can be calculated independently of its neighbors, using only data defined at its vertices, and the domain for each triangular patch can be chosen without regarding the connectivity of the patch with others. This last feature could be of great interest to a designer of a surface since one can choose the domain of each patch to be an equilateral triangle, and give it no further thought.